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Generalizations of Young's inequality
JOURNAL
OF MATHEMATICAL
ANALYSIS
Generalizations R. P. BOAS, Department
APPLICATIONS
46, 36-40
of Young’s
inequality
AND
JR. AND
M.
(1974)
B. MARCUS
of Mathematics, Northwestern Evanston, Illinois 60201
University,
1. Let f be a monotonic function whose domain contains the interval between the finite points a and b. (We do not assume that a < b.) Letf-l be a generalized inverse of f; here a generalized inverse of the nondecreasing function f is any function f -I satisfying
in+: f (x) > r> 3 f -Yy) >, su&: f (4 < r>; if f is nonincreasing, interchange inf and sup and reverse the > signs. Note that f( f -r(t)) is not necessarily equal to t. For such an f, the equation for integration by parts j-pf (4 du = bf (b) - af (4 -
i; u df (4
can be written
This is elementary when f is continuous and strictly monotonic and can be proved in the general case by approximating f by a continuous strictIy monotonic function and using bounded convergence; for a somewhat different approach compare [5, p. 1241. Two generalizations of (1) provide interesting inequalities. They are obtained by replacing f(a) or f(b) by a variable t; the equality in (1) then becomes an inequality whose sense depends on whether f is nondecreasing or nonincreasing. THEOREM. With the preceding conventions, and with the < nonincreasing functions, the > signs for nondecreasing functions, bf (4 + j--b, f -W
dr ~2 at + lef (4 4 a
af (a) + Ib f (u) du Z bt + j-;‘a’ .f-Yr> n 36 Copyright 0 1974 by Academic Press,Inc. AU rights of reproduction in any form reserved.
signs for
(2) dy.
(3)
YOUNG’S
37
INEQUALITY
Here t is in the domain off -I, but not necessarily between a and b. There is equality in (2) if and only ;f t is between f (a+) andf (a-), and in (3) if and only if t is between f (b-) and f (b+). In the first place, (2) and (3) are equivalent because we can interchange a and b; in the second place, the inequalities with decreasing functions are equivalent to those with increasing functions by the change of variable u = a + b - o. Hence, the content of the theorem is implicit in any of the four inequalities, but it is convenient to have them all. Inequality (3) for increasing functions is Young’s inequality in the form originally given by Young [6] (who considered only strictly increasing differentiable functions); the usual statement ([3, p. 1111; [4, p. 481) has a = f (a) = 0 and b > a, i.e., bt < If(u)
du + j+(y)
dy
(4)
0
and f strictly increasing. Generalized inverses were introduced in this context in [2]. The object of this note is to reintroduce Young’s original inequality, to show that it is one of four generalizations of (1) when f is monotone, and to give some applications. Although all the inequalities can be obtained from Young’s inequality, they have consequences that do not follow directly from Young’s inequality itself. Of particular interest are the inequalities for decreasing f since we can let b -+ co in (2) or a - 0 in (3) even when f (0) = co. Thus, for example, when J” f (u) du converges, bf (b) ---f 0 as b + 00 and consequently o*f-l(~) 1
dy < at + jamf(u)
4
(5)
in particular, J’. f -l(y) dy converges if J” f (u) du converges. The converse follows similarly from (3) (holding a fixed and letting t + 0). Both (2) and (3) are geometrically obvious from figures; the easiest case, the corollary of (5) just stated, simply says that the area under the graph of y = f (x) is finite if and only if the area between the graph and the y-axis is finite. We shall give an analytic proof (which seems to us simpler than the usual proofs of Young’s inequality). Finally, we shall illustrate the theorem by examples. Our experience is that such examples can be established by other methods, once found; but the generalizations of Young’s inequality are useful for finding them in the first place. 2. Proof of the Theorem. We prove (2) for decreasing functions. The proof is similar for the other cases; or, as noted above, we can derive them all from this case by simple transformations.
38
BOAS,
When
JR.
AND
MARCUS
f is nonincreasing, we obviously have (7 - 4f @I G J?f (4 du (1
(6)
whether r > a or Y < a, as long as the interval between a and r is in the domain off ; and there is equality when and only when T = a or f is constant between a and r. We can write (6) in the form rf(r)
- jbrf(s)
ds < af(r)
+ jbf(4 a
du.
Now apply integration by parts, in the form (l), to the integral on the left of (7). We get bf(b) - j.;;‘f
-‘(u)
du d af(y) + j’f@)
a
If t is in the range off, we can take r so that f(r)
du.
(8)
= t, and (8) becomes
u < at + I abf (4 du,
bf (4 + s,:, f -l(u)
with equality if and only if t = f (a). In particular, we have establishedthe “decreasing” case of (2) when f is strictly decreasing and, consequently, Young’s inequality in the usual form. We now turn to the general casewhen t is in the domain off-l but not necessarily in the range off. Unless t is between f (a+) and f(c), we can choosey. # a so that y0 is in the range off and either r,, > a, f(ro) > t, or r0 < a, f (y,,) < t. We consider the first case; the other is exactly parallel. We can then write (8) in the form bf (b) + c,
f -‘(u)
du + jt’l”‘f
-l(u)
du < af (ro) + jb f (u) du, a
I.e., bf(b)
+ J;:,,f-‘(4
du d jbf(4 a
du + af(yo) -
j;(io)f-l(u)
du.
Since t a between u = t and u = f (To), we have fb,)
st
f-‘(u)
du > 4f
(To) - t);
substituting this in (lo), we obtain (9) with strict inequality.
(10)
YOUNG’S
39
INEQUALITY
When t is between f(u+) and f(a-), (9), with since f-‘(u) = a for u between f(a) and t.
equality,
follows
from
(1)
We give some illustrations to show the advantages of 3. Illustrations. having all four inequalities (2), (3) instead of being restricted to Young’s inequality in the usual form. (a) From (5), we read off without verges because s: eex d.r converges. (b)
If we apply (5) tof(x) ab > p-w
A standard
application
calculation
= xl/+-l),
p-1 + q-1 = 1.
+ q-lb”,
+ q-‘bQ,
dr con-
0 < p < 1, we get
of (4) ([4, p. 501) to f(x)
ab < p-W
that Ji log(l/y)
p-l+
(11)
= xv-l, 4 > 1, is q-1 = 1,
(12)
from which Holder’s inequality follows for p > 1. Similarly, (11) implies Holder’s inequality for 0 < p < 1. Of course, (11) can be derived from (12), but (5) leads to (11) more directly. (c)
If we takef(t)
= et and use (2) for increasing
t - t log t < ea - at,
-ax
functions t>O.
we get (13)
This is given in [4, p. 491 (but only for a > 1) and in [3, p. 611 but with more complicated proof. If we use (5) with f(u) = e?, we get
t - t log t < e-a + at,
--cr,
t>O,
which is the same as (13) if we replace a by --a, something do on the basis of Young’s inequality (4) alone.
a
(14)
that we could not
Note that (14) says more than (13) when t < u-l sinh a. Note also that if we put t = eU and a - u = x, (13) becomes ex 2 1 + x, as was already noticed by Young [7]. (d) variable,
If we apply (2) and (5) to f(x) for 0 < u < co,
2
= e-%‘, we get, after a change of
13 e@’ dx, m s2ed ds < aemu’ + su sa ebu2 du < be-u’ + 2
u s2edS2ds, f0
O
O
40
BOAS,
(e)
Applications
A case of (5) in which
are given
JR. AND
generalized
MARCUS
inverses are essential is
in [l].
REFERENCES 1. R. P. BOAS, JR. AND M. B. MARCUS, Inequalities involving a function and its inverse, SIAM J. Math. Anal. Appl. 4 (1973), 585-591. 2. F. CUNNINGHAM, JR. AND N. GROSSMAN, On Young’s inequality, Amer. Math. Monthly 78 (1971), 781-783. 3. G. H. HARDY, J. E. LITTLEWOOD, AND G. POLYP, “Inequalities,” Cambridge University Press, 1934. Springer-Verlag, New York, 1970. 4. D. S. MITRINOVK?, “Analytic Inequalities,” 5. F. RIESZ AND B. Sz. NAGY, “Functional Analysis,” Frederick Ungar, New York, 1955. 6. W. H. YOUNG, On classes of summable functions and their Fourier series, Proc. Roy. Sot. Sec. A 87 (1912), 225-229. 7. W. H. YOUNG, On a certain series of Fourier, Proc. London Math. Sot. (2) 11 (1913), 357-366.
OF MATHEMATICAL
ANALYSIS
Generalizations R. P. BOAS, Department
APPLICATIONS
46, 36-40
of Young’s
inequality
AND
JR. AND
M.
(1974)
B. MARCUS
of Mathematics, Northwestern Evanston, Illinois 60201
University,
1. Let f be a monotonic function whose domain contains the interval between the finite points a and b. (We do not assume that a < b.) Letf-l be a generalized inverse of f; here a generalized inverse of the nondecreasing function f is any function f -I satisfying
in+: f (x) > r> 3 f -Yy) >, su&: f (4 < r>; if f is nonincreasing, interchange inf and sup and reverse the > signs. Note that f( f -r(t)) is not necessarily equal to t. For such an f, the equation for integration by parts j-pf (4 du = bf (b) - af (4 -
i; u df (4
can be written
This is elementary when f is continuous and strictly monotonic and can be proved in the general case by approximating f by a continuous strictIy monotonic function and using bounded convergence; for a somewhat different approach compare [5, p. 1241. Two generalizations of (1) provide interesting inequalities. They are obtained by replacing f(a) or f(b) by a variable t; the equality in (1) then becomes an inequality whose sense depends on whether f is nondecreasing or nonincreasing. THEOREM. With the preceding conventions, and with the < nonincreasing functions, the > signs for nondecreasing functions, bf (4 + j--b, f -W
dr ~2 at + lef (4 4 a
af (a) + Ib f (u) du Z bt + j-;‘a’ .f-Yr> n 36 Copyright 0 1974 by Academic Press,Inc. AU rights of reproduction in any form reserved.
signs for
(2) dy.
(3)
YOUNG’S
37
INEQUALITY
Here t is in the domain off -I, but not necessarily between a and b. There is equality in (2) if and only ;f t is between f (a+) andf (a-), and in (3) if and only if t is between f (b-) and f (b+). In the first place, (2) and (3) are equivalent because we can interchange a and b; in the second place, the inequalities with decreasing functions are equivalent to those with increasing functions by the change of variable u = a + b - o. Hence, the content of the theorem is implicit in any of the four inequalities, but it is convenient to have them all. Inequality (3) for increasing functions is Young’s inequality in the form originally given by Young [6] (who considered only strictly increasing differentiable functions); the usual statement ([3, p. 1111; [4, p. 481) has a = f (a) = 0 and b > a, i.e., bt < If(u)
du + j+(y)
dy
(4)
0
and f strictly increasing. Generalized inverses were introduced in this context in [2]. The object of this note is to reintroduce Young’s original inequality, to show that it is one of four generalizations of (1) when f is monotone, and to give some applications. Although all the inequalities can be obtained from Young’s inequality, they have consequences that do not follow directly from Young’s inequality itself. Of particular interest are the inequalities for decreasing f since we can let b -+ co in (2) or a - 0 in (3) even when f (0) = co. Thus, for example, when J” f (u) du converges, bf (b) ---f 0 as b + 00 and consequently o*f-l(~) 1
dy < at + jamf(u)
4
(5)
in particular, J’. f -l(y) dy converges if J” f (u) du converges. The converse follows similarly from (3) (holding a fixed and letting t + 0). Both (2) and (3) are geometrically obvious from figures; the easiest case, the corollary of (5) just stated, simply says that the area under the graph of y = f (x) is finite if and only if the area between the graph and the y-axis is finite. We shall give an analytic proof (which seems to us simpler than the usual proofs of Young’s inequality). Finally, we shall illustrate the theorem by examples. Our experience is that such examples can be established by other methods, once found; but the generalizations of Young’s inequality are useful for finding them in the first place. 2. Proof of the Theorem. We prove (2) for decreasing functions. The proof is similar for the other cases; or, as noted above, we can derive them all from this case by simple transformations.
38
BOAS,
When
JR.
AND
MARCUS
f is nonincreasing, we obviously have (7 - 4f @I G J?f (4 du (1
(6)
whether r > a or Y < a, as long as the interval between a and r is in the domain off ; and there is equality when and only when T = a or f is constant between a and r. We can write (6) in the form rf(r)
- jbrf(s)
ds < af(r)
+ jbf(4 a
du.
Now apply integration by parts, in the form (l), to the integral on the left of (7). We get bf(b) - j.;;‘f
-‘(u)
du d af(y) + j’f@)
a
If t is in the range off, we can take r so that f(r)
du.
(8)
= t, and (8) becomes
u < at + I abf (4 du,
bf (4 + s,:, f -l(u)
with equality if and only if t = f (a). In particular, we have establishedthe “decreasing” case of (2) when f is strictly decreasing and, consequently, Young’s inequality in the usual form. We now turn to the general casewhen t is in the domain off-l but not necessarily in the range off. Unless t is between f (a+) and f(c), we can choosey. # a so that y0 is in the range off and either r,, > a, f(ro) > t, or r0 < a, f (y,,) < t. We consider the first case; the other is exactly parallel. We can then write (8) in the form bf (b) + c,
f -‘(u)
du + jt’l”‘f
-l(u)
du < af (ro) + jb f (u) du, a
I.e., bf(b)
+ J;:,,f-‘(4
du d jbf(4 a
du + af(yo) -
j;(io)f-l(u)
du.
Since t
st
f-‘(u)
du > 4f
(To) - t);
substituting this in (lo), we obtain (9) with strict inequality.
(10)
YOUNG’S
39
INEQUALITY
When t is between f(u+) and f(a-), (9), with since f-‘(u) = a for u between f(a) and t.
equality,
follows
from
(1)
We give some illustrations to show the advantages of 3. Illustrations. having all four inequalities (2), (3) instead of being restricted to Young’s inequality in the usual form. (a) From (5), we read off without verges because s: eex d.r converges. (b)
If we apply (5) tof(x) ab > p-w
A standard
application
calculation
= xl/+-l),
p-1 + q-1 = 1.
+ q-lb”,
+ q-‘bQ,
dr con-
0 < p < 1, we get
of (4) ([4, p. 501) to f(x)
ab < p-W
that Ji log(l/y)
p-l+
(11)
= xv-l, 4 > 1, is q-1 = 1,
(12)
from which Holder’s inequality follows for p > 1. Similarly, (11) implies Holder’s inequality for 0 < p < 1. Of course, (11) can be derived from (12), but (5) leads to (11) more directly. (c)
If we takef(t)
= et and use (2) for increasing
t - t log t < ea - at,
-ax
functions t>O.
we get (13)
This is given in [4, p. 491 (but only for a > 1) and in [3, p. 611 but with more complicated proof. If we use (5) with f(u) = e?, we get
t - t log t < e-a + at,
--cr,
t>O,
which is the same as (13) if we replace a by --a, something do on the basis of Young’s inequality (4) alone.
a
(14)
that we could not
Note that (14) says more than (13) when t < u-l sinh a. Note also that if we put t = eU and a - u = x, (13) becomes ex 2 1 + x, as was already noticed by Young [7]. (d) variable,
If we apply (2) and (5) to f(x) for 0 < u < co,
2
= e-%‘, we get, after a change of
13 e@’ dx, m s2ed ds < aemu’ + su sa ebu2 du < be-u’ + 2
u s2edS2ds, f0
O
O
40
BOAS,
(e)
Applications
A case of (5) in which
are given
JR. AND
generalized
MARCUS
inverses are essential is
in [l].
REFERENCES 1. R. P. BOAS, JR. AND M. B. MARCUS, Inequalities involving a function and its inverse, SIAM J. Math. Anal. Appl. 4 (1973), 585-591. 2. F. CUNNINGHAM, JR. AND N. GROSSMAN, On Young’s inequality, Amer. Math. Monthly 78 (1971), 781-783. 3. G. H. HARDY, J. E. LITTLEWOOD, AND G. POLYP, “Inequalities,” Cambridge University Press, 1934. Springer-Verlag, New York, 1970. 4. D. S. MITRINOVK?, “Analytic Inequalities,” 5. F. RIESZ AND B. Sz. NAGY, “Functional Analysis,” Frederick Ungar, New York, 1955. 6. W. H. YOUNG, On classes of summable functions and their Fourier series, Proc. Roy. Sot. Sec. A 87 (1912), 225-229. 7. W. H. YOUNG, On a certain series of Fourier, Proc. London Math. Sot. (2) 11 (1913), 357-366.